This document proposes HistoSketch, a method for sketching streaming histograms that preserves similarity and adapts to concept drift. It works by: 1) Generating weighted samples from histograms such that the probability two sketches match equals histogram similarity. 2) Incrementally updating sketches using a weight decay factor to forget older data and adapt to drift over time. 3) Evaluating HistoSketch on classification tasks involving synthetic and real-world streaming data, finding it approximates histogram similarity well using small, fixed-size sketches while adapting rapidly to drift.

1. HistoSketch: Fast Similarity-Preserving Sketching
of Streaming Histograms with Concept Drift
Dingqi Yang*, Bin Li†, Laura Rettig*, Philippe Cudré-Mauroux*
*eXascale Infolab, University of Fribourg, Switzerland
†School of Computer Science, Fudan University, Shanghai, China
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2. HistoSketch: Fast Similarity-Preserving Sketching of Streaming
Histograms with Concept Drift
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What kind of location is this?
Places I’ve been:
Bar University
Museum Supermarket
0.7 0.6 0.14 0.21 0.41 0.63
0.64 0.65 0.21 0.86 0.24 0.82
0.64 0.65 0.21 0.86 0.24 0.82
0.7 0.6 0.14 0.21 0.41 0.63
Compute similarity
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3. Motivation
• Histogram similarity: foundation for many machine learning tasks
• Cardinality of histograms over data streams continuously increases
• Similarity-preserving data sketches
• Compact, fixed size
• Preserve similarity under certain measure
• Are incrementally updateable
• Concept drift: distribution of a histogram changes over time
• If taken into account can improve accuracy of histogram-based similarity
techniques
• Typical method: gradual forgetting
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4. Background
Given a data stream of incoming elements xt, with a weight wt
we compute a histogram V such that
Vi is the weighted cumulative count of the element i.
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xtxt-1...
Streaming histogram elements xt with
wt
Corresponding histogram V
xt-2

5. Problem Formulation
• Create and maintain the similarity-preserving sketch S for the full
streaming histogram V such that
• each sketch has a fixed size K (K≪ |ℰ|);
• the collision probability between two sketches Sa and Sb is the normalized
similarity between the histograms Va and Vb
 the Hamming distance between Sa and Sb approximates SIMNMM(Va, Vb);
• the sketch S(t+1) can be efficiently computed from the incoming histograms
element xt+1, S(t), and a weight decay factor λ.
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xtxt-1...
New element xt+1 received Incremental updating
xt+1
S(t+1)
xt+1S(t) λ
xt-2

6. HistoSketch
• Based on the idea of consistent weighted sampling
• Generate samples such that the probability of drawing identical samples from
two vectors is equal to their min-max similarity.
• Method draws three random variables 𝑟𝑖,𝑗~𝐺𝑎𝑚𝑚𝑎(2,1), 𝑐𝑖,𝑗~𝐺𝑎𝑚𝑚𝑎 2,1 ,
𝛽𝑖,𝑗~𝑈𝑛𝑖𝑓𝑜𝑟𝑚(0,1) and then computes
𝑦𝑖,𝑗 = exp 𝑟𝑖,𝑗
log 𝑉𝑖
𝑟𝑖,𝑗
+ 𝛽𝑖,𝑗 − 𝛽𝑖,𝑗
which is used as input to the random hash value generation.
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7. HistoSketch
• We propose a new method to compute 𝑦𝑖,𝑗
𝑦𝑖,𝑗 = exp(log 𝑉𝑖 − 𝑟𝑖,𝑗 𝛽𝑖,𝑗)
• and show that this method is 1) correct and 2) scale-invariant.
Sketch creation
𝑎𝑖,𝑗 =
𝑐𝑖,𝑗
𝑦𝑖,𝑗exp(𝑟𝑖,𝑗)
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Sketch element Sj
Histogram V
0.7 0.6 0.14 0.21 0.41ai,j
3
0.14
The corresponding
hash value Aj
Computing hash values
1 2 3 4 5i =
Minimum
1. compute 𝑦𝑖,𝑗
2. compute
hash value 𝑎𝑖,𝑗
3. set 𝑆𝑗 = 𝑎𝑟𝑔𝑚𝑖𝑛 𝑖∈ℇ
𝑎𝑖,𝑗
4. set 𝐴𝑗 = 𝑚𝑖𝑛 𝑖∈ℇ 𝑎𝑖,𝑗

8. HistoSketch
Incremental Sketch Update
Computation of sketch 𝑆 𝑡 + 1 relies only on 𝑆(𝑡) (with its corresponding hash values
𝐴 𝑡 ), an incoming element 𝑥𝑡+1 and the weight decay factor 𝜆.
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Sketch element Sj(t) 3
0.147Adjusted hash value Aj(t)e-λ
1 2 3 4 5i =
Step II. Add xt+1
- 0.142 - - -ai,j
Computing hash value for i
1 2 3 4 5i =
Adjusting sketch
Sketch element Sj(t+1)2
0.142 Hash value Aj(t+1)
Step I. Scale V(t) by e-λ
Step III. Update sketch
0.14Original hash value Aj(t)
0.14×1/(e-λ)
Minimum
1. scale existing
elements in A
2. add 𝑖′ to
histogram
3. recompute 𝑎𝑖′
,𝑗
4. update sketch 𝑆𝑗 and hash
values 𝐴𝑗 with minimum 𝑎𝑗

9. Experimental Evaluation
• Classification task
• Given labeled streaming histograms, classify those histogram instances
without label
• KNN classifier takes data in the form of sketches for classification with 𝐾 = 5
• KNN takes most up-to-date training data for classification from continuously
updated sketches
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10. Experimental Evaluation
• Synthetic dataset
• Generated from two Gaussian distributions representing two classes
• Simulate data streams with concept drift
• Abrupt: one stream starts to receive all elements from the other distribution
• Gradual: one stream starts to receive elements from the other distribution with
increasing probability, and the labels change
• Criteria:
1. How well is the similarity approximated? (impact of sketch length K)
2. How fast can it adapt to concept drift? (impact of weight decay factor λ)
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11. Experimental Evaluation
1. Impact of sketch length K
• Fix 𝜆 = 0.02 and vary
𝐾 = [20, 50, 100, 200, 500, 1000]
• Compare against two methods that
retain the full histograms:
• Histogram-Classical with unweighted
elements
• Histogram-Forgetting with gradual
forgetting weights
• A sketch length of 𝐾 = 500 is
sufficient to approximate Histogram-
Forgetting
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12. Experimental Evaluation
2. Impact of weight decay factor λ
• Fix 𝐾 = 100 and vary
𝜆 = [0, 0.005, 0.01, 0.02, 0.05, 0.1]
• Compare against Histogram-LatestK
which builds a histogram from the
latest 𝐾 = 100 elements in the
stream (unweighted)
• Similarity computation time:
• HistoSketch: 13ms
• Histogram-LatestK: 133ms
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13. Experimental Evaluation
• POI dataset
• Infer a place’s category from its customers’ visiting pattern
• Foursquare dataset: user check-ins for two years from NYC, TKY, IST
• Data: user-time visit pairs discretized to the 168 hours in a week
• Comparised methods:
• Histogram-Coarse: discretized time slots are considered as histogram elements
• Histogram-Fine-Classical: user-time pairs are considered as histogram elements
• Histogram-Fine-LatestK: only latest K histogram elements
• Histogram-Fine-Forgetting: gradual forgetting weights (𝜆 = 0.01)
• POISketch: unweighted sketching method that approximates Histogram-Fine-Classical
• HistoSketch: approximates Histogram-Fine-Forgetting (𝜆 = 0.01)
• Fix 𝐾 = 100
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14. Experimental Evaluation
Classification accuracy
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15. Experimental Evaluation
Runtime performance: classification time
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16. Conclusion
• We introduced HistoSketch, an efficient similarity preserving sketching method
for streaming histograms with concept drift.
• We demonstrated the effectiveness in approximating normalized min-max
similarity.
• We use incremental updates to the sketches with gradual forgetting to adapt to
concept drift.
• We showed on both synthetic and real-world data sets that this method
effectively and efficiently approximates similarity and adapts to concept drift.
• We observed a speed-up of 7500x on classification with a small loss of accuracy
of around 3.5%.
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Thank you!

17. Backup: Histogram Similarity
• Min-max similarity
𝑆𝑖𝑚 𝑀𝑀 𝑉 𝑎
, 𝑉 𝑏
=
Σ𝑖∈ℰmin(𝑉𝑖
𝑎
, 𝑉𝑖
𝑏
)
Σ𝑖∈ℰmax(𝑉𝑖
𝑎
, 𝑉𝑖
𝑏
)
• …normalized: sum-to-one normalization
𝑖∈ℰ
𝑉𝑖
𝑎
= 1,
𝑖∈ℰ
𝑉𝑖
𝑏
= 1
• The collision probability between two sketches 𝑆 𝑎, 𝑆 𝑏 is exactly the
normalized min-max similarity between 𝑉 𝑎, 𝑉 𝑏
Pr 𝑆𝑗
𝑎
= 𝑆𝑗
𝑏
= 𝑆𝑖𝑚 𝑁𝑀𝑀 𝑉 𝑎, 𝑉 𝑏
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18. Backup: HistoSketch Implementation
• Former histogram 𝑉 𝑡 is required to compute 𝑉(𝑡 + 1)
• The previous histogram is maintained in a modified count-min sketch 𝑄
• We extend the count-min sketch with decay weights by scaling all counters
𝑄(𝑡) ∙ 𝑒−𝜆
• Parameter configuration: 𝑑 = 10, 𝑔 = 50
guarantees an error of at most 4% with probability 0.999
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19. Backup: Experimental Evaluation
Classification accuracy over time
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20. Backup: Future Work
• Way to compute 𝑎𝑖,𝑗 can be further simplified
• Applications to other domains: e.g., recommendation, community
detection
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